Nontrivial on-site soliton solutions for stationary cubic-quintic discrete nonlinear schrodinger equation

This paper discusses the stationary cubic-quintic discrete non linear Schrodinger (CQ-DNLS) equation. The solution of this equation is determined by using the Trust-region dogleg method and the obtained solution is called a soliton. In this paper we only focus on solitons with characteristics in the form of on-site which means soliton peaked and centered at one site. In order to get the desired solution, it is necessary to choose the initial value u ∈ R^2N+1 whose characteristics are similar to on-site soliton and it is also important to choose an appropriate parameter value. Therefore, simulations were carried out by choosing parameter values w;C ∈ R. Based on the exact calculation, on-site soliton can be obtained by selecting C = 0. In case C ≠ 0, on-site soliton is obtained by using the Trust-region dogleg method, selecting the initial value u_n = sech(n), and choosing the parameter values for w and C in the intervals 0:1 ≤ w ≤ 0:41 and 0:16 ≤ C ≤ 9:3. There are differences in the shape of the on-site solitons regarding to the choices of parameter values; the greater the value of C is, the wider the soliton in the middle is. Also, the greater the value of w is, the higher the amplitude of the produced soliton is. For some parameter values, a comparison is made between the soliton generated by Trust-region dogleg and Newton method.